Second-order necessary conditions in semismooth optimization

First- and second-order conditions are given which are necessary for a functionf to have a local minimal value atx^* inRⁿ. It is assumed thatf is locally Lipschitzian nearx^* and semismooth atx^*. The necessary conditions are expressed in terms of the generalized gradients of nonsmooth analysis and certain second-order directional derivatives. The method of proof bears no resemblance to standard methods. Three special cases are discussed here, but applications to constrained problems are made elsewhere.